has the property that postcomposition with exp(2πi∫S1[S1,cconn])\exp(2 \pi i \int_{S^1}[S^1, \mathbf{c}_{conn}]) modulates the WZW 2-bundle. This is precisely the content of the second line in the table above.

Here we should regard 𝔤\mathfrak{g} as being graded and homogeneously of degree (−1)(-1) (this is the natural grading on 𝔤\mathfrak{g} regarded as an L-infinity algebra. In fact essentially all of the discussion here goes through for general L∞L_\infty-algebras equipped with a binary invariant polynomial). With this the evident total grading on ΩΣ•(−,𝔤)\Omega^\bullet_\Sigma(-,\mathfrak{g}) is already the correct BV-BRST grading

The renormalized quantum master equation

Next we want to add to the above free field theory the interaction term II. This amounts to changing the differentialQ+ℏΔQ + \hbar \Delta of ObsfreeqObs^q_{free} to Q+{I,−}+ℏΔQ + \{I,-\} + \hbar \Delta. For this indeed to still be a differential it must still square to 0, which is the condition expressed by the quantum master equation, This needs renormalization in order to be well defined.

Said heuristically: the Jones polynomial of the knot KK can be understood as the “average value” over all connections of the trace of the holonomy of the connection around the knot KK. Note that this idea can be generalized by varying the gauge group GG from SU(2)SU(2) to some other Lie group; the representation in which the trace is evaluated can also be altered. Each of these modifications gives rise to a knot invariant.

As an extended TQFT

The beautiful thing about Chern–Simons theory is that Witten was able use the locality property of the path integral to give a nonperturbative way to actually compute it. In this way Chern–Simons theory has become the ‘poster-child’ of extended topological quantum field theory since it exemplifies the main idea: take advantage of the higher gluing laws in order to compute geometric quantities.

A closed 2-manifold Σ\Sigma↦\mapsto the space of sections of the line bundle over the moduli space of flat connections on Σ\Sigma (a finite-dimensional vector space). (Reshetikhin and Turaev give an alternate quantum-groupy description of this space).

The R-T construction sticks on the circle the modular tensor category of representations of a quantum group at a root of unity, modulo “unphysical representations.” Are these supposed to be the same? Is this just the Kazhdan-Lusztig equivalence?

Urs Schreiber: the Reshetikhin-Turaev construction works with any modular tensor category, I’d say. Using one coming from reps of loops group is expected to produce the Chern–Simons QFT as a cobordism rep. But I think a full proof of that, i.e. a formalization of the CS path integral that would after turning the crank yield the RT construction, is not available to date. There is just lots of “circumstancial evidence”.

Beware that there is a subtlety in the definition of the configuration space: when the field of gravity is identified with an Iso(2,1)Iso(2,1)-connection then the configuration space naturally contains degenerate vielbein fields EE (notably the 0 vielbein) and hence the induced rank-2 tensorg=⟨E⊗E⟩g = \langle E \otimes E\rangle may also be degenerate. Such degenerate tensors are not technically pseudo-Riemannian metric tensors, since these are required to be non-degenerate. The genuine non-degenerate metric tensors correspond to precisely those Iso(2,1)Iso(2,1)-principal connections which are in fact (O(2,1)↪Iso(2,1))(O(2,1) \hookrightarrow Iso(2,1))-Cartan connections.

However, the quantum theory exists nicely if one allows the larger configuration space of possibly degenerate metrics exists nicely, while the constrained one does not. This may be interpreted as saying that at least for purposes of quantum gravity it is wrong require non-degenerate metric tensors.

Holographic relation to 2d Wess-Zumino-Witten model

Chern–Simons theory and modular forms

Trying to interest your number theory friends with Chern–Simons theory? How about this: the Chern–Simons path integralZ(k)Z(k) above is (in a certain precise sense) a modular form. This correspondence between the Chern–Simons quantum invariants and modular forms sheds light in both directions, and is a fascinating idea to me. The key words here (which I don’t understand) are “Eichler integral” and “mock theta function?”. See:

Hikami, Quantum invariant, modular forms, and lattice points, arXiv. See also the follow ups to this paper.

The Morse theory of Chern–Simons theory

In a recent talk, Witten outlined a new approach to Chern–Simons theory which perhaps gives an alternative nonperturbative definition of the path integral. Quoting from Not Even Wrong:

The main new idea that Witten was using was that the contributions of different critical points p (including complex ones), could be calculated by choosing appropriate contours 𝒞p\mathcal{C}_p using Morse theory for the Chern–Simons functional. This sort of Morse theory involving holomorphic Morse functions gets used in mathematics in Picard-Lefshetz theory?. The contour is given by the downward flow from the critical point, and the flow equation turns out to be a variant of the self-duality equation that Witten had previously encountered in his work with Kapustin on geometric Langlands. One tricky aspect of all this is that the contours one needs to integrate over are sums of the 𝒞p\mathcal{C}_p with integral coefficients and these coefficients jump at “Stokes curves” as one varies the parameter in one’s integral (in this case, x=k/nx=k/n, kk and nn are large). In his talk, Witten showed the answer that he gets for the case of the figure-eight knot.

For slides of Witten’s talk, click here and for video, click here. Pilfering material from the slides, the basic idea is as follows. Consider a general oscillatory integral in nn dimensions:

We want to make sense of the integral when the function ff is allowed to take on imaginary values (naively, the integral diverges). To do this, we use Morse theory: we choose as our Morse function the real part of the exponent, that is h=ℜ(iλf)h = \Re(i \lambda f). For every critical point pp of hh, the descending manifold CpC_p is an nn-cycle in the relative homology group Hn(X,X≪0)H_n (X, X_{\ll0}). (Basically this means that it’s a new “contour” for the integral). Moreover Morse theory tells us that the cycles we obtain in this way form a basis for the homology, so we can express our original cycle CC (the ℝn\mathbb{R}^n appearing in the integral over ℝn\mathbb{R}^n) as a linear combination of these Morse theory cycles:

(6)C=∑pnpCp
C = \sum_p n_p C_p

In this way we can make sense of the integral II by {\em defining} it as the integral over these new cycles (“contours”):

This new definition actually converges, and makes sense. Apparantly the same technique can be used to interpret the Chern–Simons path integral in the case of complex kk. Witten argues that this viewpoint is useful if we try to interpret Chern–Simons theory as a theory of three-dimensional gravity,

Which is to say What 3/4-dimensional structure is Khovanov homology hinting at? I’m inclined to think there must be one, as it seems that all of the knot homologies associated by Chern–Simons theory to representations have categorifications (I have a mostly finished paper on this). Presumably these all glue together into something, possibly by a similar trick to the Reshetikhin-Turaev construction of 3-manifold invariants, but it’s not so easy for me to see how.

Realization in physical systems

Chern-Simons theory has mostly been studied as a test case example for (pre-)quantum field theories in theoretical physics and mathematics. Also in string theory it appears in various incarnations and governs the hypothetical physics of string, notably through its holographic relation to the WZW model and the higher dimensional generalizations of this.

In the case when H•(ℰ)H^\bullet(\mathcal{E}), Kontsevich 94 and Axelrod-Singer 92, 94 (when dimM=3dim M = 3) have already constructed the perturbative Chern-Simons invariants. In some sense, their construction is orthogonal to the construction in this paper. Because we work modulo constants, the construction in this paper doesn’t give anything in the case when H•(ℰ)=0H^\bullet(\mathcal{E}) = 0. On the other hand, their constructions don’t apply in the situations where our construction gives something non-trivial. There seems to be no fundamental reason why a generalisation of the construction to this paper, including the constant term, should not exist. Such a generalisation would also generalise the results of Kontsevich and Axelrod-Singer. However, the problem of constructing such a generalisation does not seem to be amenable to the techniques used in this paper.

The BV-formalism for Chern-Simons theory on manifolds with boundary is discussed in

There is a consensus that perturbative quantization of the classical Chern-Simons theory gives the same asymptotical expansions as the combinatorial topological field theory based on quantized universal enveloping algebras at roots of unity (45), or, equivalently, on the modular category corresponding to the Wess-Zumino-Witten conformal field theory (56, 42) with the first semiclassical computations involving torsion made in (56). However this conjecture is still open despite a number of important results in this direction, see for example (47, 3).

One of the reasons why the conjecture is still open is that for manifolds with boundary the perturbative quantization of Chern-Simons theory has not been developed yet. On the other hand, for closed manifolds the perturbation theory involving Feynman diagrams was developed in (32, 27, 7) and in (5, 35, 13). For the latest development see (19). Closing this gap and developing the perturbative quantization of Chern-Simons theory for manifolds with boundary is one of the main motivations for the project started in this paper.